A quick definition check: Would someone please tell me what "locally Euclidean" means when applied to a Riemannian metric? I have kind of an idea about it, for example when we multiply by a constant... But I would like to know the more general meaning/definition of the term. Thanks.
Added: In particular, I would like to know what form such a R metric would look like e.g. $ds^2=adx^2+bdy^2$
P.S. I have tried searching for a definition, but I couldn't find a proper definition.